reserve X for set,
        n,m,k for Nat,
        K for Field,
        f for n-element real-valued FinSequence,
        M for Matrix of n,m,F_Real;

theorem Th25:
  for V be RealNormSpace,
      W be finite-dimensional RealLinearSpace,
      b be OrdBasis of RLSp2RVSp(W)
   st V is finite-dimensional
    & dim V <> 0
    & the RLSStruct of V = the RLSStruct of W
  holds
    ex k1,k2 be Real
    st 0 < k1 & 0 < k2
     & for x be Point of V
       holds
           ||.x.|| <= k1 * max_norm(W,b).x
         & max_norm(W,b).x <= k2 * ||.x.||
  proof
    let V be RealNormSpace,
        W be finite-dimensional RealLinearSpace,
        b be OrdBasis of RLSp2RVSp (W);

    assume that
    A1: V is finite-dimensional & dim V <> 0
          and
    A2: the RLSStruct of V = the RLSStruct of W;

    A4: (Omega).V = the RLSStruct of V by RLSUB_1:def 4
     .= (Omega).W by A2,RLSUB_1:def 4;

    A5: dim V
     = dim((Omega).V) by A1,RLVECT_5:30
    .= dim W by A4,RLVECT_5:30;

    reconsider e = b as FinSequence of W;
    reconsider e1 = e as FinSequence of V by A2;
    deffunc F3(Nat) = In(||.e1/.$1.||,REAL);

    consider k be FinSequence of REAL such that
    A6: len k = len b
      & for i be Nat
         st i in dom k
        holds k.i = F3(i) from FINSEQ_2:sch 1;
    set k1 = Sum(k);

    for i be Nat st i in dom k holds 0 <= k.i
    proof
      let i be Nat;
      assume i in dom k;
      then
      k.i = In(||.e1/.i.||,REAL) by A6
         .= ||.e1/.i.||;

      hence 0 <= k.i;
    end;
    then
    A7: 0 <= k1 by RVSUM_1:84;

    A8: for x be Point of V
        holds ||.x.|| <= (k1 + 1) * max_norm(W,b).x
    proof
      let x0 be Point of V;
      reconsider x = x0 as Point of W by A2;

      consider y be Element of RLSp2RVSp(W),
               z be Element of REAL(dim W) such that
      A9: x = y
        & z = y |-- b
        & max_norm(W,b).x = (max_norm(dim W)).z by Def3;

      reconsider yb = y |-- b as FinSequence of REAL;

      A10:
      now let i be Nat;
        assume
        A11: i in dom(yb);
        then i in dom(abs(z)) by A9,VALUED_1:def 11;

        then abs(z).i = |.z.i.| by VALUED_1:def 11
        .= |.yb/.i.| by A9,A11,PARTFUN1:def 6;

        hence |.yb/.i.| <= max_norm(W,b).x by A1,A5,A9,A11,Def1;
      end;

      A12: len(y |-- b) = len b by MATRLIN:def 7;
      then
      dom(y |-- b)
       = Seg len b by FINSEQ_1:def 3
      .= dom b by FINSEQ_1:def 3;

      then
      A13: dom(lmlt(y |-- b,b))
       = dom(y |-- b) by MATRLIN:12
      .= Seg len b by A12,FINSEQ_1:def 3;

      reconsider f = (the Mult of W) .: (yb,e)
        as FinSequence of the carrier of W;
      reconsider f1 = f as FinSequence of V by A2;

      A14: x = Sum(lmlt(y |-- b,b)) by A9,MATRLIN:35
      .= Sum f1 by A2;
      A15: len f = len b by A13,FINSEQ_1:def 3;

      deffunc F4(Nat) = In(||.f1/.$1.||,REAL);

      consider g be FinSequence of REAL such that
      A16: len g = len f
         & for i be Nat st i in dom g
           holds g . i = F4(i) from FINSEQ_2:sch 1;

      A17: for i be Element of NAT st i in dom f1
           holds g.i = ||. f1 /. i .||
      proof
        let i be Element of NAT;
        assume i in dom f1;
        then i in Seg(len f) by FINSEQ_1:def 3;
        then i in dom g by A16,FINSEQ_1:def 3;
        then g.i = In(||.f1/.i.||,REAL) by A16;
        hence thesis;
      end;
      then
      A18: ||.x0.|| <= Sum (g) by NDIFF_5:7,A14,A16;

      A19: for i be Nat st i in Seg len b
           holds g.i <= ((max_norm(W,b).x)*k).i
      proof
        let i be Nat;
        assume
        A20: i in Seg len b;
        A21: dom e = Seg(len b) by FINSEQ_1:def 3;
        A22: dom yb = Seg(len b) by A12,FINSEQ_1:def 3; then
        A23: yb.i = yb/.i by PARTFUN1:def 6,A20;

        f1/. i
         = f1.i by A13,A20,PARTFUN1:def 6
        .= (the Mult of W).(yb.i,e.i) by A13,A20,FUNCOP_1:22
        .= (yb/.i) * (e1/.i) by A2,A20,A21,A23,PARTFUN1:def 6;
        then
        A24: g.i
         = ||.(yb/.i) * (e1/.i) .|| by A13,A20,A17
        .= |.(yb/.i).| * ||.e1/.i.|| by NORMSP_1:def 1;

        A25: |.(yb/.i).|* ||.e1/.i.||
          <= (max_norm(W,b).x) * ||.e1/.i.||
            by XREAL_1:64,A10,A20,A22;
        i in dom k by A6,A20,FINSEQ_1:def 3;

        then k.i = In(||.e1/.i.||,REAL) by A6
        .= ||.e1/.i.||;

        hence
        g.i <= ((max_norm(W,b).x) * k).i by A24,A25,RVSUM_1:45;
      end;
      A26: g is (len b)-element by A15,A16,CARD_1:def 7;
      dom((max_norm(W,b).x) * k)
       = dom k by VALUED_1:def 5
      .= Seg len b by A6,FINSEQ_1:def 3;

      then len((max_norm(W,b).x) * k) = len b by FINSEQ_1:def 3;
      then (max_norm(W,b).x) * k is (len b)-element by CARD_1:def 7;
      then Sum(g) <= Sum((max_norm(W,b).x) * k) by A19,A26,RVSUM_1:82;
      then Sum(g) <= (max_norm(W,b).x) * Sum(k) by RVSUM_1:87;
      then
      A27: ||.x0.|| <= k1 * (max_norm(W,b).x0) by A18,XXREAL_0:2;
      A28: 0 <= max_norm(W,b).x by A1,A5,Th19;

      k1 + 0 < k1 + 1 by XREAL_1:8;
      then k1*(max_norm(W,b).x0) <= (k1+1)*(max_norm(W,b).x0)
            by A28,XREAL_1:64;
      hence ||.x0.|| <= (k1+1)*(max_norm(W,b).x0)
            by A27,XXREAL_0:2;
    end;

    consider S0 be LinearOperator of W, REAL-NS(dim W) such that
    A29: S0 is bijective
       & for x be Element of RLSp2RVSp(W)
         holds S0.x = x |-- b by A1,A5,Th23;

    reconsider S = S0
      as Function of the carrier of V, the carrier of REAL-NS (dim W) by A2;

    for x, y be Element of V
    holds S.(x + y) = S.x + S.y
    proof
      let x, y be Element of V;
      reconsider x0 = x, y0 = y as Element of W by A2;
      A30: x + y = x0 + y0 by A2;
      S0 is additive;
      hence S.(x + y) = S.x + S.y by A30;
    end;
    then
    A32: S is additive;

    for a be Real,
        x be VECTOR of V
    holds S.(a * x) = a * S.x
    proof
      let a be Real;
      let x be VECTOR of V;
      reconsider x0 = x as Element of W by A2;
      a * x = a * x0 by A2;
      hence S.(a * x) = a * S.x by LOPBAN_1:def 5;
    end;
    then S is homogeneous;
    then reconsider S as LinearOperator of V,REAL-NS (dim W) by A32;

    consider T be LinearOperator of REAL-NS (dim W),V such that
    A34: T = S " & T is one-to-one onto by A29,REAL_NS2:85;

    A35: for x be Element of V
         holds ||.x.|| <= (k1+1) * ||.S.x.||
    proof
      let x0 be Element of V;
      reconsider x = x0 as Element of W by A2;
      reconsider Sx = S.x0 as Element of REAL dim(W) by REAL_NS1:def 4;

      consider x1 be Element of RLSp2RVSp (W),
               z1 be Element of REAL (dim W) such that
      A36: x = x1
         & z1 = x1 |-- b
         & (euclid_norm(W,b)).x = |.z1.| by Def5;

      (euclid_norm(W,b)).x = ||.S.x0.|| by A29,A36,REAL_NS1:1; then
      A38: (k1 + 1) * (max_norm(W,b)).x <= (k1 + 1) * ||.S.x0.||
              by A1,A5,A7,Th22,XREAL_1:64;
      ||.x0.|| <= (k1 + 1) * max_norm(W,b).x0 by A8;

      hence ||.x0.|| <= (k1 + 1) * ||.S.x0.|| by A38,XXREAL_0:2;
    end;

    for y be Element of REAL-NS(dim W)
    holds ||.T.y.|| <= (k1 + 1) * ||.y.||
    proof
      let y be Element of REAL-NS(dim W);
      the carrier of REAL-NS(dim W) = rng S by A29,FUNCT_2:def 3;
      then
      consider x be Element of the carrier of V such that
      A39: y = S.x by FUNCT_2:113;
      T.y = x by A29,A34,A39,FUNCT_2:26;
      hence ||.T.y.|| <= (k1 + 1) * ||.y.|| by A35,A39;
    end;
    then
    A41: T is Lipschitzian by A7;

    set CW = {y where y is Element of V : max_norm(W,b).y = 1 };
    set CR = {x where x is Element of REAL-NS (dim W)
               : (max_norm(dim W)).x = 1 };

    for z be object
     st z in CW
    holds z in the carrier of V
    proof
      let z be object;
      assume z in CW; then
      ex y be Element of V
      st z = y & max_norm(W,b).y = 1;
      hence z in the carrier of V;
    end;
    then reconsider CW as Subset of V by TARSKI:def 3;

    for z be object
     st z in CR
    holds z in the carrier of REAL-NS(dim W)
    proof
      let z be object;
      assume z in CR; then
      ex y be Element of REAL-NS(dim W)
      st z = y & (max_norm(dim W)).y = 1;
      hence z in the carrier of REAL-NS(dim W);
    end;
    then
    reconsider CR as Subset of REAL-NS(dim W) by TARSKI:def 3;

    REAL-NS(dim W) is non trivial by A1,A5;
    then
    consider zn be Point of REAL-NS(dim W) such that
    A42: zn <> 0.(REAL-NS(dim W));

    reconsider znn = zn as Element of REAL(dim W) by REAL_NS1:def 4;
    zn <> 0*(dim W) by A42,REAL_NS1:def 4;
    then
    A43: (max_norm(dim W)).znn <> 0 by A1,A5,Th12; then
    A44:0 < (max_norm(dim W)).zn by A1,A5,Th12;

    set yn = (1 / (max_norm(dim W)).zn) * zn;
    set a = 1 / (max_norm(dim W)).zn;
    A45: (max_norm(dim W)).yn
     = (max_norm(dim W)).(a * znn) by REAL_NS1:3
    .= |.a.| * (max_norm(dim W)).znn by A1,A5,Th12;

    |.a.| * (max_norm(dim W)).znn
     = a * (max_norm(dim W)).zn by A44,COMPLEX1:43
    .= 1 by XCMPLX_1:106,A43;
    then
    A46: yn in CR by A45;

    A47: for y be object holds y in T.:CR iff y in CW
    proof
      let y be object;
      hereby
        assume y in T.:CR; then
        consider x be object such that
        A48: x in dom T & x in CR
           & y = T.x by FUNCT_1:def 6;

        reconsider x as Element of REAL-NS(dim W) by A48;
        A49: ex x0 be Element of REAL-NS(dim W)
             st x = x0 & (max_norm(dim W)).x0 = 1 by A48;

        consider w be Element of RLSp2RVSp(W),
                 z be Element of REAL(dim W) such that
        A50: T . x = w
           & z = w |-- b
           & max_norm(W,b).(T.x) = (max_norm(dim W)).z by A2,Def3;
        w in the carrier of V by A50;
        then
        A51: w in dom S by FUNCT_2:def 1;

        S.w = z by A29,A50;
        then
        A52: T.z = w by A29,A34,A51,FUNCT_1:34;
        z in REAL(dim W); then
        A53: z in the carrier of REAL-NS(dim W) by REAL_NS1:def 4;
        dom T = the carrier of REAL-NS(dim W) by FUNCT_2:def 1;
        then z = x by A34,A50,A52,A53,FUNCT_1:def 4;
        hence y in CW by A48,A49,A50;
      end;
      assume
      A54: y in CW;
      then reconsider y0 = y as Element of V;

      A55: ex y0 be Element of V
           st y = y0 & max_norm(W,b).y0 = 1 by A54;

      consider w be Element of RLSp2RVSp(W),
               z be Element of REAL(dim W) such that
      A56: y0 = w
         & z = w |-- b
         & max_norm(W,b).y0 = (max_norm(dim W)).z by A2,Def3;

      w in the carrier of V by A56; then
      A57: w in dom S by FUNCT_2:def 1;
      z in REAL(dim W); then
      A58: z in the carrier of REAL-NS(dim W) by REAL_NS1:def 4;
      then
      A59: z in dom T by FUNCT_2:def 1;
      S.w = z by A29,A56;
      then
      A60: T.z = w by A29,A34,A57,FUNCT_1:34;
      z in CR by A56,A55,A58;
      hence y in T.:CR by A56,A59,A60,FUNCT_1:def 6;
    end;

    the carrier of REAL-NS(dim W) = REAL(dim W) by REAL_NS1:def 4;
    then reconsider g = max_norm(dim W)
          as Function of the carrier of REAL-NS(dim W),REAL;
    set D = the carrier of REAL-NS(dim W);

    A63: dom g = D by FUNCT_2:def 1;
    A64: |.-1 .|
     = |. 1 .| by COMPLEX1:52
    .= 1 by COMPLEX1:43;

    for x0 be Point of REAL-NS(dim W)
    for r be Real st x0 in D & 0 < r
    holds
      ex s be Real st 0 < s
        &
      for x1 be Point of REAL-NS(dim W)
      st x1 in D & ||.x1 - x0.|| < s
      holds |.((g /. x1) - (g /. x0)).| < r
    proof
      let x0 be Point of REAL-NS(dim W);
      let r be Real;
      assume
      A65: x0 in D & 0 < r;
      set s = r;
      take s;
      thus 0 < s by A65;
      let x1 be Point of REAL-NS(dim W);
      assume
      A66: x1 in D & ||.x1 - x0.|| < s;
      A67: dom g = D by FUNCT_2:def 1;

      reconsider y0 = x0 as Element of REAL(dim W)
        by REAL_NS1:def 4;
      reconsider y1 = x1 as Element of REAL(dim W) by REAL_NS1:def 4;

      A68: ||.x1 - x0.|| = |.y1 - y0.| by REAL_NS1:1,REAL_NS1:5;
      A69: (max_norm(dim W)).(y1 - y0) <= |.y1 - y0.| by A1,A5,Th14;
      A70: x1 - x0 = y1 - y0 by REAL_NS1:5;
      then
      A71: y0 + (y1 - y0) = x0 + (x1 - x0) by REAL_NS1:2
      .= y1 by RLVECT_4:1;
      A72: x0 - x1 = y0 - y1 by REAL_NS1:5; then
      A73: y1 + (y0 - y1) = x1 + (x0 - x1) by REAL_NS1:2
      .= y0 by RLVECT_4:1;

      y0 - y1
       = -(x1 - x0) by A72,RLVECT_1:33
      .= (-1) * (x1-x0) by RLVECT_1:16
      .= (-1) * (y1-y0) by A70,REAL_NS1:3;
      then
      A74: (max_norm(dim W)).(y0-y1)
       = |.-1 .| * (max_norm(dim W)).(y1-y0) by Th12,A1,A5
      .= (max_norm(dim W)).(y1-y0) by A64;

      A75: g.(y1-y0) = g/.(y1-y0) by A67,A70,PARTFUN1:def 6;
      A76: g/.y1 = g.y1 by A67,PARTFUN1:def 6;
      A77: g/.y0 = g.y0 by A67,PARTFUN1:def 6;

      A78:(max_norm(dim W)).y1 - (max_norm(dim W)).y0
       <= (max_norm(dim W)).y0 + (max_norm(dim W)).(y1-y0)
           -(max_norm(dim W)).y0 by A1,A5,A71,Th12,XREAL_1:9;

      (max_norm(dim W)).y1
        - ((max_norm(dim W)).y1 + (max_norm(dim W)).(y0-y1))
      <= (max_norm(dim W)).y1 - (max_norm(dim W)).y0
        by A1,A5,A73,Th12,XREAL_1:10; then
      - (max_norm(dim W)).(y1-y0)
      <= (max_norm(dim W)).y1 - (max_norm(dim W)).y0 by A74;
      then |. g/.y1 - g/.y0 .| <= g/.(y1 - y0) by A75,A76,A77,A78,ABSVALUE:5;
      then |. g/.x1 - g/.x0 .| <= ||.x1 - x0.|| by A68,A69,A75,XXREAL_0:2;
      hence |. g/.x1 - g/.x0 .| < s by A66,XXREAL_0:2;
    end;
    then
    A79: g is_continuous_on the carrier of REAL-NS(dim W) by A63,NFCONT_1:20;

    for s1 be sequence of REAL-NS(dim W)
     st rng s1 c= CR & s1 is convergent
    holds lim s1 in CR
    proof
      let s1 be sequence of REAL-NS (dim W);
      assume
      A80: rng s1 c= CR & s1 is convergent;
      set D = the carrier of REAL-NS(dim W);
      A82: g is_continuous_in (lim s1) by A79;
      A83: dom s1=NAT by FUNCT_2:def 1;
      A81: D = dom g by FUNCT_2:def 1; then
      A84: rng s1 c= dom g;
      then
      A85: g/*s1 = g*s1 by FUNCT_2:def 11;
      then
      A86: dom(g/*s1) = dom s1 by A84,RELAT_1:27;

      for x, y be object
      st x in dom(g/*s1) & y in dom(g/*s1)
      holds (g/*s1) . x = (g/*s1) . y
      proof
        let x, y be object;
        assume
        A87: x in dom(g/*s1) & y in dom(g/*s1);
        then reconsider i = x, j = y as Element of NAT;

        i in dom s1 by A84,A85,A87,RELAT_1:27;
        then s1.i in rng s1 by FUNCT_1:3;
        then s1.i in CR by A80;
        then
        ex x be Element of REAL-NS (dim W)
        st s1.i = x & (max_norm(dim W)).x = 1;
        then
        A88: (g/*s1).x = 1 by A84,FUNCT_2:108;

        j in dom s1 by A87,A85,A84,RELAT_1:27;
        then s1.j in rng s1 by FUNCT_1:3;
        then s1.j in CR by A80;
        then
        ex x be Element of REAL-NS (dim W)
        st s1.j = x & (max_norm(dim W)).x = 1;
        hence thesis by A84,FUNCT_2:108,A88;
      end;

      then
      A89: g/*s1 is constant by FUNCT_1:def 10;
      A90: (g/*s1).1 in rng (g/*s1)
         & (g/*s1).1 = g.(s1.1)
            by A83,A84,A86,FUNCT_1:3,FUNCT_2:108;
      s1.1 in rng s1 by A83,FUNCT_1:3;
      then s1.1 in CR by A80;
      then
      A92: ex x be Element of REAL-NS (dim W)
           st s1.1 = x & (max_norm(dim W)).x = 1;

      (max_norm(dim W)).(lim s1)
       = lim (g/*s1) by A80,A81,A82
      .= 1 by A89,A90,A92,SEQ_4:25;
      hence lim s1 in CR;
    end;

    then
    A93: CR is closed;
    A95: 0 + dim W < 1 + dim W by XREAL_1:8;
    A94: ex r be Real
         st for y be Point of REAL-NS(dim W)
             st y in CR
            holds ||.y.|| < r
    proof
      set r = 1 + dim W;
      take r;
      let y be Point of REAL-NS(dim W);
      assume y in CR;
      then
      A96: ex z be Element of REAL-NS(dim W)
           st y = z & (max_norm(dim W)).z = 1;

      reconsider y0 = y as Element of REAL(dim W) by REAL_NS1:def 4;

      (sum_norm(dim W)).y0 <= (dim W) * (max_norm(dim W)).y0
        &
      (max_norm(dim W)).y0 <= |.y0.|
        &
      |.y0.| <= (sum_norm(dim W)).y0 by A1,A5,Th14;

      then |.y0.| <= (dim W) * (max_norm(dim W)).y0 by XXREAL_0:2;
      then ||.y.|| <= (dim W) by A96,REAL_NS1:1;
      hence ||.y.|| < r by A95,XXREAL_0:2;
    end;

    A97: T is_continuous_on CR by A41,LOPBAN_5:4,NFCONT_1:23;
    CW = T.:CR by A47,TARSKI:2; then
    A98: CW is compact by A93,A94,A97,Lm3,NFCONT_1:32;

    reconsider f = id CW as PartFunc of V,V;

    dom f = CW;
    then
    A99: f is_continuous_on CW by NFCONT_1:50;

    dom T = the carrier of REAL-NS(dim W) by FUNCT_2:def 1;
    then
    A100: CW <> {} by A47,TARSKI:2,A46;
    A101: dom f = dom ||.f.|| by NORMSP_0:def 3;
    then
    A102: rng ||.f.|| is compact by A98,A99,NFCONT_1:28,NFCONT_1:31;
    rng ||.f.|| <> {} by A101,A100,RELAT_1:42;
    then consider y0 be Element of V such that
    A104: y0 in dom ||.f.||
        & lower_bound(rng ||.f.||) = ||.f.|| . y0
          by A102,PARTFUN1:3,RCOMP_1:14;
    A107: ||.f.|| . y0 = ||.f/.y0.|| by A104,NORMSP_0:def 3;
    rng ||.f.|| is real-bounded by A102,RCOMP_1:10;
    then rng ||.f.|| is bounded_below by XXREAL_2:def 11;
    then
    A105: for r be Real st r in (rng ||.f.||)
          holds ||.f.|| . y0 <= r by A104,SEQ_4:def 2;

    set k2 = ||.f/.y0.||;

    A108: for x be Element of V
           st x in CW
          holds k2 <= ||.x.||
    proof
      let x be Element of V;
      assume
      A109: x in CW; then
      f/.x = f.x by A101,PARTFUN1:def 6
      .= x by A109,FUNCT_1:18; then

      ||.f.|| . x = ||.x.|| by A101,A109,NORMSP_0:def 3;
      hence k2 <= ||.x.|| by A101,A105,A107,A109,FUNCT_1:3;
    end;

    A113: k2 <> 0
    proof
      assume k2=0; then
      consider x be Element of V such that
      A114: x in dom ||.f.|| & 0 = ||.f.||.x by A104,A107;

      ||.f.||.x = ||.f/.x.|| by NORMSP_0:def 3,A114;
      then f/.x = 0.V by A114,NORMSP_0:def 5;
      then f.x = 0.V by A101,A114,PARTFUN1:def 6;
      then 0.V in CW by A101,A114,FUNCT_1:18;
      then
      A115: ex y be Element of V st y= 0.V & max_norm(W,b).y = 1;

      max_norm(W,b).(0.V)
       = max_norm(W,b).(0.W) by A2
      .= 0 by A1,A5,Th19;
      hence contradiction by A115;
    end;
    then
    A116: 0 < 1/k2 by XREAL_1:139;
    for x be Point of V
    holds max_norm(W,b).x <= (1/k2)*||.x.||
    proof
      let x be Element of V;
      reconsider xn = x as Element of W by A2;
      per cases;
      suppose
        x = 0.V; then
        max_norm(W,b).x
         = max_norm(W,b).(0.W) by A2
        .= 0 by A1,A5,Th19;
        hence max_norm(W,b).x <= (1/k2)*||.x.||;
      end;

      suppose
        x<> 0.V;
        then xn <> 0.W by A2;
        then
        A118: (max_norm(W,b)).xn <> 0 by A1,A5,Th19; then
        A119: 0 < (max_norm(W,b)).xn by A1,A5,Th19;

        set y = (1/max_norm(W,b).x) * x;
        set a = 1/(max_norm(W,b)).x;
        a * x = a * xn by A2;
        then
        A120: (max_norm(W,b)).y = |.a.| *(max_norm(W,b)).xn by A1,A5,Th19;

        |.a.| * (max_norm(W,b)).xn
         = a* (max_norm(W,b)).xn by A119,COMPLEX1:43
        .= 1 by A118,XCMPLX_1:106; then
        y in CW by A120; then
        A123: k2 * (max_norm(W,b).x) <= ||.y.|| * (max_norm(W,b).x)
              by A108,XREAL_1:64,A119;
        ||.y.|| = |.1 / (max_norm(W,b).x).| * ||.x.|| by NORMSP_1:def 1
        .= 1 / (max_norm(W,b).x) * ||.x.|| by A119,COMPLEX1:43; then

        ||.y.|| * (max_norm(W,b).x)
         = (1/ (max_norm(W,b).x))*(max_norm(W,b).x) *||.x.||
        .= 1 * ||.x.|| by A118,XCMPLX_1:106
        .= ||.x.||;

        then (1/k2) * (k2 * (max_norm(W,b).x)) <= (1/k2) * ||.x.||
          by A123,XREAL_1:64;
        then (1/k2) * k2 * ((max_norm(W,b)).x) <= (1/k2) * ||.x.||;
        then 1 * (max_norm(W,b).x) <= (1/k2) * ||.x.||
          by A113,XCMPLX_1:106;
        hence (max_norm(W,b).x) <= (1/k2)*||.x.||;
      end;
    end;
    hence thesis by A7,A8,A116;
  end;
